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Program Topical Lectures December 2007

Program Topical Lectures December 2007. CP Violation in the Standard Model. Topical Lectures Nikhef Dec 12, 2007 Marcel Merk. Part 1: Introduction: Discrete Symmetries Part 2: The origin of CP Violation in the Standard Model Part 3: Flavour mixing with B decays

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Program Topical Lectures December 2007

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  1. Program TopicalLectures December 2007

  2. CP Violation in the Standard Model Topical Lectures Nikhef Dec 12, 2007 Marcel Merk Part 1: Introduction: Discrete Symmetries Part 2: The origin of CP Violation in the Standard Model Part 3: Flavour mixing with B decays Part 4: Observing CP violation in B decays

  3. Apologies… Sometimes I can’tresistshowingsomesillypic’s…

  4. Preliminary remarks • The lectures are aimed at graduate students who are non-experts in flavour physics and CP violation. • Apologies for those who are active in the field. • I will focus on concepts rather than on state of the art measurements and exact theoretical derivations. • “Undemocratic” presentation: • Almost none of the beautiful kaon physics will be discussed. • Very much looking forward to these sessions…

  5. Questions are Welcome (1)

  6. Questions are Welcome (2) Answers might depend on the context…

  7. Literature References: • C.Jarlskog, “Introduction to CP Violation”, Advanced Series on Directions in High Energy Physics – Vol 3: “CP Violation”, 1998, p3. • Y.Nir, “CP Violation In and Beyond the Standard Model”, Lectures given at the XXVII SLAC Summer Institute, hep-ph/9911321. • Branco, Lavoura, Silva: “CP Violation”, International series of monographs on physics, Oxford univ. press, 1999. • Bigi and Sanda: “CP Violation”, Cambridge monographs on particle physics, nuclear physics and cosmology, Cambridge univ. press, 2000. • T.D. Lee, “Particle Physics and Introduction to Field Theory”, Contemporary Concepts in Physics Volume 1, Revised and Updated First Edition, Harwood Academic Publishers, 1990. • C. Quigg, “Gauge Theories of the Strong, Weak and Electromagnetic Interactions”, Frontiers in Physics, Benjamin-Cummings, 1983.

  8. Introduction: Symmetry and non-Observables T.D.Lee: “The root to all symmetry principles lies in the assumption that it is impossible to observe certain basic quantities; the non-observables” • There are four main types of symmetry: • Permutation symmetry: • Bose-Einstein and Fermi-Dirac Statistics • Continuous space-time symmetries: • translation, rotation, acceleration,… • Discrete symmetries: • space inversion, time inversion, charge inversion • Unitary symmetries: gauge invariances: • U1(charge), SU2(isospin), SU3(color),.. • If a quantity is fundamentally non-observable it is related to an exact symmetry • If a quantity could in principle be observed by an improved measurement; the symmetry is said to be broken conservation law Noether Theorem: symmetry

  9. Symmetry and non-observables Simple Example: Potential energy V between two charged particles: Absolute position is a non-observable: The interaction is independent on the choice of the origin 0. Symmetry: V is invariant under arbitrary space translations: 0’ 0 Consequently: Total momentum is conserved:

  10. Symmetry and non-observables

  11. Puzzling thought… (to me, at least) COBE: • Can we use the “dipole asymmetry” in cosmic microwave background to define an absolute Lorentz frame in the universe? • If so, what does it imply for Lorentz invariance? WMAP:

  12. Three Discrete Symmetries +  • Parity, P unobs.: (absolute handedness) • Parity reflects a system through the origin. Convertsright-handed coordinate systems to left-handed ones. • Vectors change sign but axial vectors remain unchanged • x  -x , p  -p, butL = x  p  L • Charge Conjugation, C unobs.: (absolute charge) • Charge conjugation turns a particle into its anti-particle • e +e- , K -K + • Time Reversal, T unobs.: (direction of time) • Changes, for example, the direction of motion of particles • t -t

  13. Parity The parity operation performs a reflection of the space coordinates at the origin: • If we apply the parity operation to a wave function , we get another wave function ’ with: which means that P is a unitary operation. • If P =a , then  is an eigenstate of parity, with eigenvaluea. For example: The combination  = cosx+sinx is not an eigenstate of P Spin-statistics theorem: bosons(1,2)  +(2,1) symmetric fermions(1,2)  -(2,1) antisymmetric

  14. Parity • One can apply the parity operation to physical quantities: • Mass m P m = mscalar • Force FP F(x) = F(-x) = -F(x)vector • Acceleration aP a(x) = a(-x) = -a(x)vector • It follows that Newton’s law is invariant under the parity operation • There are also vectors which do not change sign under parity. They are usually derived from the cross product of two other vectors, e.g. the magnetic field: • These are called axial vectors. • Finally, there are also scalar quantities which do change sign under the parity operation. They are usually an inner product of a vector and a axial vector, e.g. the electric dipole moment (s is the spin): These are the pseudoscalars.

  15. Charge conjugation • The charge conjugation C is an operation which changes the charge (and all other internal quantum numbers). Applied to the Lorentz force it gives: which shows that this law is invariant under the C operation. • Generally, when applied to a particle, the charge conjugation inverts the charge and the magnetic moment of a particle leaving other quantities (mass, spin, etc.) unchanged. • Only neutral states can be eigenstates, e.g. Evidently, and so C isunitary, too.

  16. C and P operators In Diractheoryparticles are representedbyDiracspinors: Antimatter! +1/2, -1/2 helicity solutionsfor the particle +1/2, -1/2 helicity solutionsfor the antiparticle Implementation of the P and C conjugation operators in DiracTheory is (See H&M section 5.4 and 5.6) However: In general C and P are onlydefined up to phase, e.g.: Note: quantumnumbersassociatedwith discrete operations C and P are multiplicative in contrast to quantumnumbersassociatedbycontinuoussymmetries

  17. Time reversal • Time reversal is analogous to the parity operation, except that the time coordinate is affected, not the space coordinate • Again the macroscopic laws of physics are unchanged under the operation of time reversal(although some people find it hard to imagine the time inverse of a broken mirror…),the law remains invariant since t appears quadratically. • Other vectors, like momentum and velocity, which are linear motions, change sign under time reversal. So do the magnetic field and spin, which are due to the motion of charge.

  18. Time reversal: antiunitary Wigner found that T operator is antiunitary: • This leaves the physical content of a system unchanged, since: • Antiunitary operators may be interpreted as the product of an unitary operator by an operator which complex-conjugates. • As a consequence, T is antilinear: • Consider time reversal of the free Schrodinger equation: Complex conjugation is required to stay invariant under time reversal

  19. C-,P-,T-, Symmetry • The basic question of Charge, Parity and Time symmetry can be addressed as follows: • Suppose we are watching some physical event. Can we determine unambiguously whether: • we are watching this event in a mirror or not? • Macroscopic asymmetries are considered to be accidents on life’s evolution rather then a fundamental asymmetry of the laws of physics. • we are watching the event in a film running backwards in time or not? • The arrow of time is due to thermodynamics: i.e. the realization of a macroscopic final state is statistically more probably than the initial state. • It is not assigned to a time-reversal asymmetry in the laws of physics. • we are watching the event where all charges have been reversed or not? • Classical Theory (Newton mechanics, Maxwell Electrodynamics) are invariant under C,P,T operations, i.e. they conserve C,P,T symmetry

  20. Macroscopic time reversal(T.D. Lee) • At eachcrossing: 50% - 50% choice to go leftor right • Aftermanydecisions: invert the velocity of the final state and return • Do we end up with the initial state?

  21. Macroscopic time reversal(T.D. Lee) Veryunlikely! • At eachcrossing: 50% - 50% choice to go leftor right • Aftermanydecisions: invert the velocity of the final state and return • Do we end up with the initial state?

  22. Parity Violation Before 1956 physicists were convinced that the laws of nature were left-right symmetric. Strange? A “gedanken” experiment: Consider two perfectly mirror symmetric cars: Gas pedal Gas pedal driver driver “L” and “R” are fully symmetric, Each nut, bolt, molecule etc. However the engine is a black box “R” “L” Person “L” gets in, starts, ….. 60 km/h Person “R” gets in, starts, ….. What happens? What happens in case the ignition mechanism uses, say, Co60b decay?

  23. Parity Violation! e- Parity transformation q q   J J Magnetic field 60Co 60Co B e-  More electrons emitted opposite the J direction. Not random -> Parity violation! • Sketch and photograph of apparatus used to study beta decay in polarized cobalt-60 nuclei. The specimen, a cerium magnesium nitrate crystal containing a thin surface layer of radioactive cobalt-60, was supported in a cerium magnesium nitrate housing within an evacuated glass vessel (lower half of photograph). An anthracene crystal about 2 cm above the cobalt-60 source served as a scintillation counter for beta-ray detection. Lucite rod (upper half of photograph) transmitted flashes from the counter to a photomultiplier (not shown). Magnet on either side of the specimen was used to cool it to approximately 0.003 K by adiabatic demagnetization. Inductance coil is part of a magnetic thermometer for determining specimen temperature.

  24. Weak Force breaks C and P, is CP really OK ? C e-R W- nL P e+L e-L W+ W- nR nR e+R W+ nL • Weak Interaction breaks both C and P symmetry maximally! • Despite the maximal violation of C and P symmetry, the combined operation, CP, seemed exactly conserved… • But, in 1964, Christensen, Cronin, Fitch and Turlay observed CP violation in decays of Neutral Kaons!

  25. Testing CP conservation q K2p+p- Effect is tiny: about 2/1000 Create a pure KL (CP=-1) beam: (Cronin & Fitch in 1964) Easy: just “wait” until the Ks component has decayed… If CP conserved, should not see the decay KL→ 2 pions Main background: KL->p+p-p0 … and for this experiment they got the Nobel price in 1980…

  26. Contact with Aliens ! Are they made of matter or anti-matter?

  27. Contact with Aliens ! CPLEAR, Phys.Rep. 374(2003) 165-270 Compare the charge of the most abundantly produced electron with that of the electrons in your body: If equal: anti-matter If opposite: matter

  28. CPT Invariance • Local Field theories always respect: • Lorentz Invariance • Symmetry under CPT operation (an electron = a positron travelling back in time) • => Consequence: mass of particle = mass of anti-particle: (Lüders, Pauli, Schwinger) (anti-unitarity) => Similarly the total decay-rate of a particle is equal to that of the anti-particle • Question 1: • The mass difference between KL and KS: Dm = 3.5 x 10-6eV => CPT violation? • Question 2: • How come the lifetime of KS = 0.089 ns while the lifetime of the KL = 51.7 ns? • Question 3: • BaBar measures decay rate B→J/y KS and B→J/y KS. Clearly not the same: how can it be? Answer 1 + 2: A KL≠ an anti-KSparticle! Answer 3: Partial decay rate ≠ total decay rate! However, the sum over all partial rates (>200 or so) is the same for B and B. (Amazing! – at least to me)

  29. CPT Violation…

  30. CP Violation in the Standard Model Topical Lectures Nikhef Dec 12, 2007 Marcel Merk Part 1: Introduction: Discrete Symmetries Part 2: The origin of CP Violation in the Standard Model Part 3: Flavour mixing with B decays Part 4: Observing CP violation in B decays

  31. CP in the Standard Model Lagrangian(The origin of the CKM-matrix)

  32. CP in the Standard Model Lagrangian(The origin of the CKM-matrix) • Plan: • Look at symmetry aspects of the Lagrangian • How is CP violation implemented? → Several “miracles” happen in symmetry breaking LSM contains: LKinetic : fermion fields LHiggs: the Higgs potential LYukawa: the Higgs – fermion interactions Standard Model gauge symmetry: Note Immediately: The weak part is explicitly parity violating • Outline: • Lorentz structure of the Lagrangian • Introduce the fermion fields in the SM • LKinetic : local gauge invariance : fermions ↔ bosons • LHiggs : spontaneous symmetry breaking • LYukawa: the origin of fermion masses • VCKM : CP violation

  33. Lagrangian Density Local field theories work with Lagrangian densities: with the fields taken at The fundamental quantity, when discussing symmetries is the Action: If the action is (is not) invariant under a symmetry operation then the symmetry in question is a good (broken) one => Unitarity of the interaction requires the Lagrangian to be Hermitian

  34. Structure of a Lagrangian Lorentz structure: interactions can be implemented using combinations of: S: Scalar currents : 1 P: Pseudoscalarcurrents : g5 V: Vector currents : gm A: Axial vector currents : gmg5 T: Tensor currents : smn Dirac field  : Scalar field f: Example: Consider a spin-1/2 (Dirac) particle (“nucleon”) interacting with a spin-0 (Scalar) object (“meson”) Nucleon field Meson potential Nucleon – meson interaction Exercise: What are the symmetries of this theory under C, P, CP ? Can a and bbe any complex numbers? Note: the interaction term contains scalar and pseudoscalar parts Violates P, conserves C, violates CP a and b must be real from Hermeticity

  35. Transformation Properties (Ignoring arbitrary phases) Transformation properties of Dirac spinorbilinears (interaction terms): c→c* c→c*

  36. The Standard Model Lagrangian • LKinetic: •Introduce the masslessfermion fields • •Require local gauge invariance => gives rise to existence of gauge bosons => CP Conserving • LHiggs: •Introduce Higgs potential with <f> ≠ 0 • •Spontaneous symmetry breaking The W+, W-,Z0 bosons acquire a mass => CP Conserving • LYukawa: •Ad hoc interactions between Higgs field & fermions => CP violating with a single phase • LYukawa→ Lmass: • fermion weak eigenstates: • -- mass matrix is (3x3) non-diagonal • • fermion mass eigenstates: • -- mass matrix is (3x3) diagonal => CP-violating => CP-conserving! • LKineticin mass eigenstates: CKM – matrix => CP violating with a single phase

  37. Fields: Notation Interaction rep. Y SU(3)C SU(2)L Left- handed generation index Q = T3 + Y Fermions: with y = QL, uR, dR, LL, lR, nR Quarks: Under SU2: Left handed doublets Right hander singlets Leptons: Note: Interaction representation: standard model interaction is independent of generation number Scalar field:

  38. Fields: Notation Q = T3 + Y Explicitly: • The left handed quark doublet : • Similarly for the quark singlets: • The left handed leptons: • And similarly the (charged) singlets:

  39. Intermezzo: Local Gauge Invariance in a single transparancy Basic principle: The Lagrangian must be invariant under local gauge transformations Example: massless Dirac Spinors in QED: “global” U(1) gauge transformation: “local” U(1) gauge transformation: Is the Lagrangian invariant? Not invariant! Then: Then it turns out that: => Introduce the covariant derivative: and demand that Am transforms as: is invariant! • Introduce charged fermion field (electron) • Demand invariance under local gauge transformations (U(1)) • The price to pay is that a gauge field Am must be introduced at the same time (the photon) Conclusion:

  40. :The Kinetic Part Fermions + gauge bosons + interactions Procedure: Introduce the Fermion fields and demand that the theory is local gauge invariant under transformations. Start with the Dirac Lagrangian: Replace: Gam :8 gluons Wbm: weak bosons: W1, W2, W3 Bm: hyperchargeboson Fields: Generators: La : Gell-Mann matrices: ½ la(3x3) SU(3)C Tb : Pauli Matrices: ½ tb(2x2) SU(2)L Y : Hypercharge: U(1)Y For the remainder we only consider Electroweak: SU(2)L x U(1)Y

  41. : The Kinetic Part uLI W+m g dLI Exercise: Show that this Lagrangian formally violates both P and C Show that this Lagrangian conserves CP LKin = CP conserving For example the term with QLiIbecomes: and similarly for all other terms (uRiI,dRiI,LLiI,lRiI). Writing out only the weak part for the quarks: W+ = (1/√2) (W1+ i W2) W- = (1/√ 2) (W1– i W2) L=JmWm

  42. : The Higgs Potential • . • The W+,W-,Z0 bosons acquire mass • The Higgs boson H appears →Note LHiggs = CP conserving V(f) V(f) Symmetry Broken Symmetry f f ~ 246 GeV Spontaneous Symmetry Breaking: The Higgs field adopts a non-zero vacuum expectation value Procedure: Substitute: And rewrite the Lagrangian (tedious): (The other 3 Higgs fields are “eaten” by the W, Z bosons) “The realization of the vacuum breaks the symmetry”

  43. : The Yukawa Part Since we have a Higgs field we can add (ad-hoc) interactions between f and the fermions in a gauge invariant way. doublets L must be Her- mitian (unitary) The result is: singlet With: (The C-conjugate of f To be manifestly invariant under SU(2) ) are arbitrary complex matrices which operate in family space (3x3) => Flavour physics!

  44. : The Yukawa Part Writing the first term explicitly: For For the nucleonpotential we had aninteraction term: Question: In what aspect is this Lagrangian similar to the example of the nucleon-meson potential?

  45. : The Yukawa Part Formally, CP is violated if: In general LYukawa is CP violating Exercise (intuitive proof) Show that: • The hermiticity of the Lagrangian implies that there are terms in pairs of the form: • However a transformation under CP gives: CP is conserved in LYukawaonly if Yij = Yij* and leaves the coefficientsYij and Yij*unchanged

  46. : The Yukawa Part There are 3 Yukawa matrices (in the case of massless neutrino’s): • Each matrix is 3x3 complex: • 27 real parameters • 27 imaginary parameters (“phases”) • many of the parameters are equivalent, since the physics described by one set of couplings is the same as another • It can be shown (see ref. [Nir]) that the independent parameters are: • 12 real parameters • 1 imaginary phase • This single phase is the source of all CP violation in the Standard Model ……Revisit later

  47. So far, so good…?

  48. Hope not…

  49. : The Fermion Masses S.S.B Start with the Yukawa Lagrangian n is vacuumexpectationvalue of the Higgspotential After which the following mass term emerges: with LMassis CP violating in a similar way as LYuk

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